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Resource-efficient digital quantum simulation of $d$-level systems for photonic, vibrational, and spin-$s$ Hamiltonians

Nicolas P. D. Sawaya, Tim Menke, Thi Ha Kyaw, Sonika Johri, Alán Aspuru-Guzik, Gian Giacomo Guerreschi

arXiv:1909.12847v3quant-ph

TL;DR

Qubit-based simulation of d-level systems requires selecting encodings under qubit and gate constraints. This paper compares encoding strategies and conversions, finding that the optimal choice depends on Hamiltonian class and truncation level d.

  • Problem

    Resource requirements for qubit-based simulation of d-level systems involve selecting encodings under constraints on both qubit count and gate count.

    Source: §I · pp. 1, 3

  • Method

    The paper develops a framework for encoding d-level systems and compares qubit and entangling-operation requirements across encodings for composite Hamiltonians, including conversion costs.

    Source: §I · p. 17, §III · p. 18

  • Results

    The optimal encoding and whether to interconvert depend on Hamiltonian class and truncation level d; Gray tends to suit tridiagonal operators, while SB tends to suit a common diagonal class.

    Source: §III · p. 18

  • Takeaways & Limitations

    Choosing a simulation strategy requires weighing operation counts against conversion costs and qubit-memory constraints.

    Source: §III · p. 18

    The resource counts assume all-to-all connectivity; accounting for hardware topology and SWAP gates could change the counts and some observed trends.

    Source: §III · p. 18

Abstract

from arXiv · show

Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin-$\frac{1}{2}$ systems. Here, we instead consider encodings of $d$-level (i.e. qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin-$s$ and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been used in this context previously. After outlining a methodology for implementing an arbitrary encoding, we investigate the interplay between Hamming distances, sparsity patterns, bosonic truncation, and other properties of local operators. Finally, we obtain resource counts for five common Hamiltonian classes used in physics and chemistry, while modeling the possibility of converting between encodings within a Trotter step. The most efficient encoding choice is heavily dependent on the application and highly sensitive to $d$, although clear trends are present. These operation count reductions are relevant for running algorithms on near-term quantum hardware because the savings effectively decrease the required circuit depth. Results and procedures outlined in this work may be useful for simulating a broad class of Hamiltonians on qubit-based digital quantum computers.

I. INTRODUCTION · II. RESULTS · A. Binary encodings

The paper develops and compares qubit encodings for d-level systems, finding that the best choice depends on the Hamiltonian, truncation, and hardware resource limits. It defines compact, unary, and block-unary encodings with distinct resource and transition properties.

I. INTRODUCTION

  • Qudits appear in bosonic, vibrational, spin-s, and molecular systems, and this work develops a flexible mapping framework to compare encodings for operators and Hamiltonians.The framework is intended to help identify efficient mappings for particular operators and hardware, including NISQ devices.Source: §I · pp. 1, 3
  • Encoding choices must meet both qubit and gate budgets, so an encoding that works for one hardware profile may fail for another.Figure 1 illustrates this trade-off for truncation d = 5: device A accommodates Gray and standard binary, while device B accommodates unary.Source: Figure 1 · p. 2, §I · p. 3
  • Across five Hamiltonian classes, optimal encodings vary with the Hamiltonian and d, sometimes favoring conversion and sometimes remaining in one encoding; Figure 2 summarizes the comparisons.Scenarios A and B avoid unary, while C favors unary with possible compaction and D favors remaining in unary throughout.Source: Figure 2 · p. 2, §I · p. 3

A. Binary encodings

  • Binary encodings map integers to ordered bit strings; standard binary uses ⌈log2 d⌉ qubits for levels 0 through d−1.The paper uses “binary encoding” broadly for integer representations as ordered binary numbers, not only base-two numbering.Source: A. Binary encodings · pp. 3–4
  • Gray code assigns adjacent integers bit strings with Hamming distance one, benefiting tridiagonal operators with zero diagonals.The authors state that, to their knowledge, Gray code had not previously been proposed for qubit-based simulation of bosons and other d-level systems.Source: A. Binary encodings · p. 4
  • Unary encoding uses d qubits, while its bitmask subset for level l is just bit l; compact standard-binary and Gray codes require all Nq bits to identify a level.Unary uses only an exponentially small subspace of the qubits’ Hilbert space.Source: A. Binary encodings · p. 4
  • Block unary interpolates between compact and unary encodings, enabling tunable qubit-operation trade-offs for hardware budgets.It combines a compact local encoding with block size g; transitions between blocks are conditional on all qubits in both blocks.Source: A. Binary encodings · p. 4

B. Mapping d-level Matrix Operators to Qubits · C. Significance of the Hamming distance

The paper maps d-level matrix operators to qubit Pauli strings through arbitrary binary encodings, then shows how Hamming distance affects Pauli-string length and implementation cost.

B. Mapping d-level Matrix Operators to Qubits

  • For bosonic modes, the mapping requires choosing a finite level cutoff, while determining a cutoff that meets an error threshold is outside this work’s scope.The levels are treated as orthonormal states, primarily in Fock-type encodings; the procedure also applies to first-quantization operators.Source: B. Mapping d-level Matrix Operators to… · pp. 4–5
  • The general procedure assigns each level an integer and encodes that integer as a qubit bitstring using an arbitrary binary code.The method represents each level |l⟩ by the encoded register |R(l)⟩.Source: B. Mapping d-level Matrix Operators to… · p. 5
  • For non-compact encodings, the mapping can ignore qubits outside the bitmask union for each matrix element, then convert the remaining local terms into Pauli strings.Each resulting Pauli string contains Pauli operators or identities, with coefficients.Source: B. Mapping d-level Matrix Operators to… · pp. 5–6

C. Significance of the Hamming distance

  • As Hamming distance between encoded levels increases, the average Pauli-string length increases; this consideration applies to compact and non-compact encodings.For non-compact codes, the relevant bits are those in the union of the two levels’ bitmask subsets.Source: C. Significance of the Hamming distance · p. 6
  • Matched bits introduce more identities into the expansion, reducing non-identity Pauli operators and therefore the quantum operations needed to implement the Hamiltonian.The passage connects this mechanism to the expansion of each encoded matrix element into Pauli strings.Source: C. Significance of the Hamming distance · p. 6
  • For |3⟩⟨4| + |4⟩⟨3|, Gray code gives Hamming distance 1 and fewer Pauli operators per string than SB, requiring fewer implementation operations.The paper illustrates the comparison using the Pauli-string representations of the same Hermitian term in each encoding.Source: C. Significance of the Hamming distance · p. 6

D. Avoiding superfluous terms in non-compact codes · E. Trotterization and gate count upper bounds

For non-compact codes, multiply operators in their matrix representation before encoding to avoid superfluous terms. Trotter-step gate upper bounds vary with encoding and matrix sparsity, with Hamming distance and bitmask size explaining key trends.

D. Avoiding superfluous terms in non-compact codes

  • For unary or block-unary codes, multiply local operators as matrices before encoding; multiplying their qubit representations can introduce superfluous terms.The resulting Pauli operators can act identically on the encoding subspace while differing in implementation cost, and eliminating extra terms after mapping may require difficult combinatorial analysis; this concern does not apply to compact codes.Source: D. Avoiding superfluous terms in… · p. 7

E. Trotterization and gate count upper bounds

  • The resource study evaluates one Suzuki-Trotter step, while leaving encoding-dependent simulation error for future work.The decomposition is exact in the limit of large η or small t; although other methods may be asymptotically more efficient, cited results suggest first-order Trotterization can have lower error on near- and medium-term hardware [39].Source: E. Trotterization and gate count upper… · p. 8
  • A Pauli string is exponentiated with a CNOT staircase circuit, with additional single-qubit basis-change gates for qubits acted on by X or Y.The CNOT count depends on p, the number of non-identity Pauli operators in the term.Source: E. Trotterization and gate count upper… · p. 8
  • The analytical entangling-gate bounds cover a single Hermitian element pair, tridiagonal B with O(d) nonzero entries, and dense operators with O(d^2) entries.Bounds depend on Hamming distance and the size of the relevant bitmask union; circuit compilation can cancel gates, limiting their direct use in encoding choice.Source: E. Trotterization and gate count upper… · p. 8
  • For O(d)-density operators, unary often has lower gate upper bounds, while compact codes can be superior for dense operators and scale no worse than O(d^2 log d).For dense operators, unary's O(d^2) upper bound is only slightly lower; actual counts may be below these bounds, so compact codes may outperform unary in both qubit and gate counts.Source: E. Trotterization and gate count upper… · pp. 8–9
  • For tridiagonal B, qubit counts and CNOT upper bounds show an approximate inverse relationship: unary and block unary use more qubits but have smaller bitmask subsets.Figure 4 reports numerical upper bounds for one Suzuki-Trotter step; its caption notes that encodings with higher qubit counts tend to have lower gate-count upper bounds.Source: E. Trotterization and gate count upper… · p. 9, Figure 4 · p. 9
  • Gray-code upper bounds are substantially lower than SB bounds for B because every nonzero term has Hamming distance one; unary bounds are asymptotically lower, with BUGray g intermediate.Unary keeps K=2 per matrix element, while Gray and SB use K=ceil(log2 d); after Pauli-term cancellation and circuit optimization, these trends are less pronounced and less predictable.Source: E. Trotterization and gate count upper… · pp. 9–10

F. Diagonal Binary-Decomposable Operators

Diagonal binary-decomposable (DBD) operators have diagonal entries that admit binary decomposition, and their structure can make exact exponentiation especially inexpensive when the dimension is a power of two.

F. Diagonal Binary-Decomposable Operators

  • DBD operators are diagonal operators whose entries admit binary decomposition; diagonal evenly-spaced (DES) operators are a subclass that includes the bosonic number operator and a n + bI.Here a and b are constants.Source: F. Diagonal Binary-Decomposable… · p. 10
  • For d a power of two, exactly implementing exp(-iθn) takes log2 d single-qubit rotations and no entangling gates; Figure 5 also shows the SB mapping is superior for d a power of two.For n^2, entangling gates are still required, though the passage reports an advantage when d is a power of two.Source: F. Diagonal Binary-Decomposable… · p. 11
  • When log2 d is not an integer, the single-qubit-only decomposition does not generally apply; for truncated bosonic n with d at least 11, using SB encoding at d = 16 is most efficient.For spin-s, S_z is DBD, but the SB advantage applies only when d = 2s+1 is a power of two, such as s = 3/2 or 7/2.Source: F. Diagonal Binary-Decomposable… · p. 11
  • The square of a DES or DBD operator is not generally DBD.Source: F. Diagonal Binary-Decomposable… · p. 11

G. Local operators

Local-operator circuit costs depend on operator sparsity, encoding, and level count: Gray often helps for tridiagonal operators, while diagonal or pentadiagonal structure changes the comparison. Bosonic truncation can also be chosen to reduce gates without adding qubits, whereas spin levels are fixed by s.

G. Local operators

  • Bosonic q is tridiagonal with nonzero transitions only between neighboring levels, whereas q^2 is pentadiagonal with nonzero entries separated by zero- and two-level offsets, making Gray’s unit Hamming distance less useful.The momentum operator p and its square share the respective sparsity patterns of q and q^2; for compact encodings, implementing q^2 at d=8 rather than truncating at d=5 can reduce gates without increasing qubits.Source: G. Local operators · pp. 11, 14
  • For bosonic operators, Gray usually beats SB on tridiagonal operators, but its advantage is weaker for q^2; at d=4, 7, and 8, Gray or SB can beat unary in both qubits and operations.The q^2 advantage changes with d: Gray is asymptotically better than SB, while SB is better at lower d; q^2 also has operation-count dips at d=8, 24, and 32 because its diagonal is DBD.Source: Figure 6 · p. 12, G. Local operators · pp. 13–14
  • The section also describes a first-quantized boson mapping in discretized position space, whose approximate position operator is DBD and can be mapped with the same procedure using Nx instead of d.The mapping discretizes harmonic-oscillator Hermite-Gauss functions; the grid spacing is chosen so the desired highest-order function lies within the grid interval.Source: G. Local operators · p. 13
  • Circuit optimization reduces compact-code and block-unary gate counts for q by factors of 2 to 3, while unary gains little but retains the fewest entangling gates for many d values.The optimized circuits were compiled by searching for and performing gate cancellations, including adjacent CNOT or Hadamard gates.Source: G. Local operators · p. 13
  • Spin-operator trends are less orderly: SB is especially efficient for diagonal Sz at s=3/2 and 7/2, while Gray usually beats SB for tridiagonal Sx and unary is often worse in both gate and qubit counts.Spin-system d is fixed by s, and Sz and the two-particle Sz interaction show similar encoding trends.Source: Figure 7 · p. 13, G. Local operators · p. 14
  • Gray-based block unary is never the sole encoding with the lowest entangling-gate count, but may suit unusual hardware budgets that trade memory against operations; the reported savings are generally constant-factor when d does not grow with system size.Different Hamiltonian terms may be Trotterized in different encodings when the cost savings exceed conversion overhead.Source: G. Local operators · p. 14

H. Conversions between encodings

The section explains why switching encodings between Hamiltonian terms can improve resource efficiency and presents conversion circuits and resource counts for the encodings considered. It also describes limits on analyzing block-unary conversions.

H. Conversions between encodings

  • Conversion costs can be substantially smaller than the resource-efficiency gap between encodings, making it advantageous to switch encodings between Hamiltonian terms.For the Bose-Hubbard model, the number operator is usually more efficient in SB, while the hopping term is usually more efficient in Gray.Source: H. Conversions between encodings · p. 14
  • Gray and SB conversion requires ⌈log2 d⌉−1 sequential CNOTs.The section shows the SB-to-Gray circuit.Source: H. Conversions between encodings · p. 14, Figure 9 · p. 15
  • For many d values, a bosonic interaction can be implemented more cheaply in unary by expanding the register, then compacting it back; savings depend on the problem and parameters.The example expands each particle from ⌈log2 d⌉ qubits to d qubits before compacting back to SB or Gray.Source: Figure 8 · p. 15
  • SB-to-unary conversion uses enlarged memory, requires no ancillas, and is reversible; general-d resource counts are provided, with modifications when d is not a power of two.The procedure includes non-power-of-two cases; example circuits for d = 5 and 7 are given in the supplementary section.Source: H. Conversions between encodings · pp. 16–17, Table IV · p. 17
  • A converter between SB and block unary was constructed for g = 3, but these conversions are not further analyzed because block unary has limited expected utility and decoherence times may be too low.Source: H. Conversions between encodings · p. 17

I. Composite Systems

The section compares five encoding strategies for five composite Hamiltonian classes and finds that the most effective strategy depends strongly on the problem and d. The novel memory-efficient schemes can reduce two-qubit gate counts relative to both unary-only and SB-only encodings.

I. Composite Systems

  • The comparisons use optimized Trotterized-operator and inter-encoding-conversion circuits across SB-only, Gray-only, unary-only, SB–Gray conversion, and three-code compaction schemes.Results are relative to SB; conversion costs are included for the combined schemes, and the reported combined SB–Gray results improve on both individual codes while the three-code results appear only when unary is best among the first four.Source: §I · p. 17
  • For selected local bosonic operators, gate counts can decrease nonmonotonically with d, so the composite-system estimates raise the cutoff when that reduces operations without using more qubits.For q^2, a sufficient truncation of d = 5 can use resource counts for d = 8; this adjustment does not apply to spin-s systems, where d is fixed by particle spin.Source: §I · p. 17
  • The best encoding can vary sharply with d, and results for d = 4 and d = 10 illustrate this variation across the shifted 1D QHO, Bose-Hubbard, Franck-Condon, spin-s Heisenberg, and boson-sampling Hamiltonians.Franck-Condon instances assume a Duschinsky matrix with k = 4 nonzero entries per row; relative resource differences are constant-factor savings independent of system size when d is constant.Source: §I · p. 17
  • Scenario A favors one compact encoding throughout, while Scenario B favors compact storage with Gray–SB conversion when different local operators favor each code.Examples include the Bose-Hubbard and 1D QHO at d = 4 for A, and the Heisenberg model at s = 7/2 and Franck-Condon at d = 4 for B; conversion cost is very small.Source: §I · pp. 17–18
  • Scenario C favors unary even with repeated compaction, whereas Scenario D favors unary only when memory permits staying unary; otherwise, use Gray and/or SB.Scenario C occurs for bosonic Bose-Hubbard, boson-sampling, and Franck-Condon cases at d = 10 despite SB-to-unary conversion costing approximately 9d entangling gates; Scenario D includes the 1D QHO at d = 10 and Heisenberg at s = 2.Source: §I · p. 18
  • Up to 33% fewer two-qubit entangling gates than memory-inefficient unary-only and up to 49% fewer than memory-efficient SB-only were observed with the novel schemes.Memory-efficient schemes store encoded subsystems in ⌈log2 d⌉ qubits except possibly during operations; the novel approaches did not improve every instance.Source: §I · p. 18

III. DISCUSSION

The paper compares qubit and entangling-operation costs for encoding d-level systems, finding that encoding efficiency depends on the operator and simulation context. Its results motivate evaluating strategies for each Hamiltonian class and truncation level.

III. DISCUSSION

  • The framework compares integer-to-bit encodings by qubit counts and entangling-operation requirements for Hamiltonians involving bosons, spin-s particles, molecular electronic levels, and vibrational modes.The mappings also apply to other d-level subsystems.Source: §III · p. 18
  • Gray tends to be more efficient than SB for tridiagonal operators, while SB tends to be superior for a common diagonal class; interconversion within Suzuki-Trotter steps often saves resources.Unary encoding often uses more qubits but fewer operations, while SB or Gray can be more efficient in both qubit and operation counts. The authors note that Gray code had not previously been used in Hamiltonian simulation.Source: §III · p. 18
  • Across five composite Hamiltonian classes, optimal encoding and interconversion depend heavily on the Hamiltonian class and particle truncation level d.The four simulation scenarios distinguish strategies by operation counts, interconversion cost, and qubit-memory constraints; observed anomalies motivate numerically evaluating each new simulation problem before running it on hardware.Source: §III · p. 18
  • Future resource estimates could include T-gate counts for error-corrected hardware and account for connectivity topology and SWAP gates, which may alter counts and observed trends.The work assumes all-to-all connectivity, which the passage identifies as typical of ion-trap quantum computers.Source: §III · p. 18
  • The authors envision their methodology and results helping theorists and experimentalists design resource-efficient simulations of physically and chemically relevant Hamiltonians.Source: §III · p. 19

FIGURE LEGENDS · IV. SB to BUSB

The figure legends show that encoding efficiency depends on the Hamiltonian, level count, and hardware constraints, and describe resource tradeoffs and conversion strategies across encodings.

FIGURE LEGENDS

  • Hardware budgets can favor different encodings: unary exceeds the qubit budget on device A, while Gray and SB exceed device B’s operation budget.The schematic concerns first-order Trotterization of q^2 for one boson with d = 5.Source: FIGURE LEGENDS · p. 21
  • The optimal encoding for first-order Trotterization varies with Hamiltonian class and level count, and sometimes benefits from converting between encodings.The comparisons cover Bose-Hubbard, one-dimensional QHO, Franck-Condon, boson sampling, and spin-s Heisenberg Hamiltonians, with costs reported relative to standard binary.Source: FIGURE LEGENDS · p. 21
  • For tridiagonal real Hermitian operators, SB and Gray use logarithmically scaling qubit counts, while unary and block unary scale linearly.The legend notes that encodings with higher qubit counts tend to have lower upper bounds for gate counts, and vice versa; the Pauli-string circuit uses 2(p −1) two-qubit gates before potential cancellation across products of exponentials.Source: FIGURE LEGENDS · p. 21
  • For diagonal evenly spaced operators, unary always uses only single-qubit operations, while SB needs no entangling gates when log2 d is an integer.In that case SB uses log2 d single-qubit operations and is most efficient in both qubits and operations; the operator class includes bosonic number and spin-Sz operators.Source: FIGURE LEGENDS · p. 21
  • Gray usually beats SB for the plotted bosonic operators, but its advantage for q^2 is reduced because that operator is pentadiagonal.Unary usually requires the fewest operations as d increases, though Gray or SB can be more efficient in both qubit and operation count at values such as d = 4, 7, and 8; reductions for q^2 at d = 8, 24, or 32 arise from its DBD diagonal.Source: FIGURE LEGENDS · pp. 21–22
  • Converting from SB or Gray to unary for an interaction and then compacting back can save operations, but whether it pays off depends on the problem and parameters.The example expands each particle from ceil(log2 d) qubits to d qubits before compacting back.Source: FIGURE LEGENDS · p. 22

I. UPPER BOUNDS FOR ENTANGLING GATES

The section derives Pauli-string-based upper bounds on entangling-gate counts and applies them to sparse and dense operators across compact, unary, and block unary encodings. For the stated sparsity pattern, unary has a lower asymptotic bound than compact codes, while the advantage diminishes for denser operators.

I. UPPER BOUNDS FOR ENTANGLING GATES

  • For Hermitian off-diagonal term pairs, Pauli-string length distributions are binomial and depend on the encoding through the Hamming distance between encoded levels.The distribution includes a factor of 1/2 because imaginary-coefficient Pauli terms cancel in the Hermitian sum.Source: §I · p. 24
  • These are upper bounds because circuit construction can allow gate cancellations, and repeated Pauli strings can combine, so circuit optimization is needed for more accurate resource estimates.The analytical expressions are intended to explain general trends rather than always identify the best encoding for a given operator.Source: §I · pp. 25–26
  • For sparsity pattern B̂ with O(d) nonzero entries, unary has an O(d) CNOT upper bound versus O(d^2 log d) for compact Gray and standard-binary encodings, at the cost of more qubits.The unary bound is O(1) per Hermitian term pair, independent of d.Source: §I · p. 25
  • For dense O(d^2)-entry operators, compact encodings have an O(d^2 log d) entangling-gate upper bound, and unary’s advantage diminishes as matrix density increases.The compact bound counts O(d^2) distinct Pauli strings, each requiring at most O(log d) CNOT gates.Source: §I · p. 25
  • For block unary, a Hermitian pair yields O(g log g) operations, giving bounds of O(dg log g) for sparsity pattern B̂ and O(d^2 log g) for dense matrices.The bitmask subset union for one pair has K = O(log g) bits.Source: §I · p. 26

II. CIRCUIT OPTIMIZATION · III. SB TO UNARY CONVERSION: GENERAL CASE · IV. SB TO BUSB

These sections describe how Pauli-term ordering and iterative gate optimization reduce circuit gate counts, then give SB-to-unary conversion details for non-power-of-two d and an SB-to-BUSB circuit example. They also note that BUSB conversion may be too costly under hardware gate-count constraints.

II. CIRCUIT OPTIMIZATION

  • CNOT counts per Suzuki-Trotter step were determined by converting d-level operators to Pauli operators, collecting coefficients, and cancelling Pauli terms with QubitOperator [MSK+17].The software packages used were FermiLib and OpenFermion [MSK+17].Source: §II · p. 26
  • Adjacent Pauli-term CNOT ladders expose canceling pairs of CNOT, Hadamard, or iRx+y(π) gates, which are eliminated as identities.Circuits were synthesized by converting each Pauli term to a CNOT ladder and placing ladders next to each other.Source: §II · p. 26
  • The optimizer repeatedly commutes gates to cancel inverses or merge rotations, applies reduction patterns, and converges over several passes.One pattern replaces CNOT(i0,i1) × CNOT(i1,i2) × CNOT(i0,i1) with CNOT(i0,i2) × CNOT(i1,i2).Source: §II · p. 26
  • The default ordering beat >900 random orderings for encoded bosonic q in unary, Gray, and standard binary codes, although finding an optimal ordering is combinatorially hard [BKM18].The orderings were tested for each code; placing similar Pauli strings side by side can create more adjacent gates that cancel.Source: §II · pp. 26–27

III. SB TO UNARY CONVERSION: GENERAL CASE

  • For non-power-of-two d, SB-to-unary conversion shifts the most-significant qubit and controls to d−1, removes unused qubits, and relocates the final CNOT target.For d=5, the binary base line shifts from 7 to 4; the final CNOT has control 4 and target 0.Source: §III · p. 27

IV. SB TO BUSB

  • A circuit demonstrates SB-to-BUSB conversion for g=3 and d=12, showing that conversion between these encodings is possible in principle.The supplementary figure depicts standard binary to block unary conversion.Source: §IV · p. 27, Figure 2 · p. 27
  • The general SB-to-BUSB circuit is omitted because rare hardware budgets where BU may help are likely to lack gate-count capacity for conversion.Source: §IV · p. 27

V. FIRST QUANTIZATION

First quantization represents bosonic degrees of freedom using first-quantized QHO operators, and QFTs allow switching between position and momentum bases. For an approximate QHO Hamiltonian, this enables a Trotter sequence of squared position and momentum operators.

V. FIRST QUANTIZATION

  • A QHO Trotter step applies the position-squared operator, QFT, momentum-squared operator, then inverse QFT to return to the position basis.First-quantized QHO operators obey bosonic algebra, so they can represent bosonic degrees of freedom.Source: §V · p. 27
  • Trotterizing the first-quantized position and momentum operators takes log2 d single-qubit rotations and no two-qubit gates, while exponentiating their squares requires entangling gates.The passage also describes the operations as scaling approximately as O(log2 d).Source: §V · p. 28
  • Initial-state preparation is a major hurdle: preparing a QHO ground state requires a normalized Gaussian first Hermite-Gauss function, with options discussed by Macridin et al. [MSAH18a].The passage says a variational approach can be more efficient than expected for preparing ground states on near- or medium-term quantum computers.Source: §V · p. 28
  • The paper expects early bosonic simulations to favor second-quantized Fock mappings because Fock-state preparation is just bit flips, though non-Fock wave-packet dynamics may benefit from first quantization.Mapping first-quantized position and momentum operators to qubits follows the general matrix-mapping procedure, replacing d with Nx.Source: §V · p. 28

VI. HAMILTONIAN DEFINITIONS

The section defines Hamiltonians for five simulation settings: lattice bosons, shifted and multidimensional vibrational systems, higher-spin magnets, and optical boson sampling. It specifies their model parameters, truncation assumptions, or gate-by-gate representation.

Bose-Hubbard model

  • The study uses a periodic one-dimensional Bose–Hubbard model, parameterized by hopping t, on-site energy U, chemical potential μ, levels d per site, and particle count N.The model is associated with Mott-insulator and superfluid phases, with additional terms enabling more complex phases.Source: §VI · p. 28

Shifted one-dimensional QHO

  • The shifted one-dimensional quantum harmonic oscillator is a bosonic-encoding prototype and models Franck–Condon factors between two electronic transitions in a diatomic molecule.Its displacement δ indirectly determines how many levels d have appreciable intensity in overlaps with the unshifted oscillator.Source: §VI · p. 28

Multidimensional Franck-Condon factors

  • Multidimensional Franck–Condon simulation treats each vibrational normal mode as a bosonic mode, with Duschinsky transformations between electronic-surface bases.The required dimension d can range from 1 to 70; resource-count simulations assume sparse S with k = 4 and equal d across modes, although real molecules may use different d values.Source: §VI · pp. 28–29

Spin-s Heisenberg model

  • The spin-s model is a one-dimensional Heisenberg chain with ferromagnetic SzSz interactions and a transverse magnetic field.Higher-spin systems are relevant to nuclear magnetic resonance and studies of magnetic materials.Source: §VI · p. 29

Boson sampling

  • Boson sampling is modeled as layer-by-layer phase-shifter and beam-splitter unitaries, whose generators yield effective Hamiltonians for first-order Trotterization on qubits.Unlike the other examples, this setting is stroboscopic rather than evolution under one large Hamiltonian.Source: §VI · pp. 29–30

VII. FULL COMPOSITE HAMILTONIAN RESULTS

The section presents two-qubit entangling-gate resource requirements relative to SB for five Hamiltonian classes, with encoding scenarios assigned by problem and parameter values. It also describes which conversion and compacting results are omitted from the plots.

VII. FULL COMPOSITE HAMILTONIAN RESULTS

  • Supplementary Figures 3–7 plot two-qubit entangling-gate resource requirements relative to SB, with bosonic Hamiltonians up to d=16 and Heisenberg up to s=7/2; captions give scenarios A–D per combination.The one-dimensional QHO runs from d=3 through 16; the Bose-Hubbard and boson-sampling plots run from d=4 through 16.Source: §VII · p. 30, Figure 3 · p. 31, Figure 5 · p. 32
  • Conversion results are omitted when they do not outperform Gray-only and SB-only, and “All with compacting” is omitted when unary-only is not better than the compact schemes.Many plotted “SB & Gray” results are only marginally (<1%) better than SB-only and Gray-only.Source: §VII · p. 30
  • Supplementary Figures 3–7 compare two-qubit entangling-gate requirements relative to SB for the one-dimensional QHO, transverse Heisenberg, Bose–Hubbard, boson-sampling, and Franck–Condon problems.The latter four cases allow arbitrary particle or mode counts.Source: §VII · p. 30, Figure 3 · p. 31, Figure 4 · p. 31

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